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On the prescribed Q-curvature problem on S n . (English) Zbl 1233.53007
Assuming n5, on a smooth compact Riemannian manifold (M n ,g), the Q-curvature is a fourth-order invariant involved in the Paneitz operator expression. The present work studies the problem of prescribing the Q-curvature on the standard n-sphere. The main assumption here is that the order β of flatness at critical points of the prescribed Q-curvature satisfies n-4β<n. The problem has a variational structure, but the lack of compactness does not permit to use standard variational methods. In particular, the problem fails to satisfy the Palais-Smale condition, which is the main difficulty of this problem. The outline of the proof is the following: first, a study of the critical points at infinity of the variational problem; second, the computation of the Morse index of each critical point at infinity; and then, a comparison of the total index with the Euler-Poincaré characteristic of the space of variation. The characterization of the critical points at infinity is obtained through the construction of a suitable pseudo-gradient at infinity. This construction is based on a very delicate expansion of the gradient of the associated Euler-Lagrange functional near infinity. New existence results are obtained through formulas of Euler-Hopf type. The proof gives an upper bound on the Morse index of the obtained solutions and a lower bound on the number of conformal metrics having the same Q-curvature.
MSC:
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
58J60Relations of PDE with special manifold structures
53C43Differential geometric aspects of harmonic maps
References:
[1]Abdelhedi, W.; Chtioui, H.: On the prescibed paneitz curvature problem on the standard spheres, Adv. nonlinear stud. 4, 511-528 (2006) · Zbl 1184.53043
[2]Bahri, A.: Critical point at infinity in some variational problems, Pitman res. Notes math. 182 (1989) · Zbl 0676.58021
[3]Bahri, A.: An invariant for yamabe-type flows with applications to scalar curvature problems in high dimensions, A celebration of J.F. Nash jr., Duke math. J. 81, 323-466 (1996) · Zbl 0856.53028 · doi:10.1215/S0012-7094-96-08116-8
[4]Bahri, A.; Coron, J. M.: The scalar curvature problem on the standard three dimensional spheres, J. funct. Anal. 95, 106-172 (1991) · Zbl 0722.53032 · doi:10.1016/0022-1236(91)90026-2
[5]Bahri, A.; Rabinowitz, P.: Periodic orbits of Hamiltonian systems of three body type, Ann. inst. H. Poincaré anal. Non linéaire 8, 561-649 (1991) · Zbl 0745.34034 · doi:numdam:AIHPC_1991__8_6_561_0
[6]Ben Ayed, M.; El Mehdi, K.: The paneitz curvature problem on lower dimensional spheres, Ann. global anal. Geom. 31, No. 1, 1-36 (2007) · Zbl 1170.35394 · doi:10.1007/s10455-005-9003-7
[7]Ben Ayed, M.; Chen, Y.; Chtioui, H.; Hammami, M.: On the prescribed scalar curvature problem on 4-manifolds, Duke math. J. 84, 633-677 (1996) · Zbl 0862.53034 · doi:10.1215/S0012-7094-96-08420-3
[8]Chang, S. Y. A.; Yang, P. C.: Extremal metrics of zeta function determinants on 4-manifolds, Ann. of math. 142, 171-212 (1995) · Zbl 0842.58011 · doi:10.2307/2118613
[9]Chtioui, H.; Rigane, A.: On the prescribed Q-curvature problem on sn, C. R. Acad. sci. Paris ser. I 348, 635-638 (2010) · Zbl 1193.53101 · doi:10.1016/j.crma.2010.03.018
[10]Djadli, Z.; Hebey, E.; Ledoux, M.: Paneitz type operators and application, Duke math. J. 104, No. 1, 129-169 (2000) · Zbl 0998.58009 · doi:10.1215/S0012-7094-00-10416-4
[11]Djadli, Z.; Malchiodi, A.: Existence of conformal metrics with constant Q-curvature, Ann. of math. (2) 168, No. 3, 813-858 (2008) · Zbl 1186.53050 · doi:10.4007/annals.2008.168.813 · doi:http://annals.math.princeton.edu/annals/2008/168-3/p03.xhtml
[12]Djadli, Z.; Malchiodi, A.; Ahmadou, M. Ould: Prescribing a fourth order conformal invariant on the standard sphere. II: blow up analysis and applications, Ann. sc. Norm. super. Pisa cl. Sci. (5) 1, 387-434 (2002) · Zbl 1150.53012
[13]Felli, V.: Existence of conformal metrics on sn with prescribed fourth-order invariant, Adv. differential equations 7, 47-76 (2002) · Zbl 1054.53061
[14]Gursky, M.: The principal eigenvalue of a conormally invariant differential operator, with an application to semilinear elliptic PDE, Comm. math. Phys. 207, 131-147 (1999) · Zbl 0988.58013 · doi:10.1007/s002200050721
[15]Hatcher, Allen: Algebraic topology, (2002)
[16]Li, Y. Y.: Prescribing scalar curvature on sn and related topics, part I, J. differential equations 120, 319-410 (1995)
[17]Li, Y. Y.: Prescribing scalar curvature on sn and related topics, part II : Existence and compactness, Comm. pure appl. Math. 49, 541-579 (1996) · Zbl 0849.53031 · doi:10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A
[18]Schoen, R.; Zhang, D.: Prescribed scalar curvature on the n-sphere, Calc. var. Partial differential equations 4, 1-25 (1996) · Zbl 0843.53037 · doi:10.1007/BF01322307
[19]Wei, J.; Xu, X.: On conformal deformations of metrics on sn, J. funct. Anal. 157, 292-325 (1998)